In addition to the standard LQC, in which the Lorentzian term of the classical Hamiltonian constraint is first expressed in terms of the Euclidean term in the spatially flat
Friedmann-Lemaître-Robertson-Walker (FLRW) universe, and then only the quantization of the Euclidean term is considered, the robustness of the singularity resolution with respect to different quantizations of the classical Hamiltonian constraint in the symmetry reduced spacetimes have been extensively studied. Two notable examples are the so-called modified LQC-I (mLQC-I) and modified LQC-II (mLQC-II) models, which were first proposed by Yang, Ding and Ma more than a decade ago [1]}.
In a recent study, Dapor and Liegener (DL) [2]}, [3]} obtained the expectation values of the Hamiltonian operator in LQG using complexifier coherent states [4]}, adapted to the spatially flat FLRW
universe. Using the non-graph changing regularization of the Hamiltonian advocated by Thiemann [5]}, DL obtained an effective Hamiltonian constraint, which, to the leading order, agrees with the mLQC-I model first obtained in [1]}. Sometimes, this model has also been referred to as the DL model or Thiemann regularized LQC. Strictly speaking,
when constructing loops in [2]} DL treated the edge length \(\mu \)  as a free parameter, but in [1]} it was considered as a specific triad dependent function, the so-called \(\bar{\mu }\)  scheme [9]}, which is known to be the only possible choice in LQC, and results in physics that is independent from underlying fiducial structures used during quantization, and meanwhile yields a consistent infrared behavior for all matter obeying the weak energy condition [10]}. Lately, the studies of [2]} have been extended to the \(\bar{\mu }\)  scheme [12]}.
